A321179 a(n) = [x^(n^2)] Product_{k=1..n} theta_3(q^k), where theta_3() is the Jacobi theta function.

1, 2, 2, 14, 44, 174, 988, 4314, 20780, 126320, 692328, 3836166, 23160914, 135752866, 803203484, 4902966108, 29745996950, 181712320506, 1124481497694, 6965802854354, 43360326335154, 271658784580760, 1706393926177980, 10757142052998054, 68081390206251952, 432001821971576352

Offset: 0

Author

Seiichi Manyama, Oct 29 2018

Keywords

• nonn

Comments

Also the number of integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n^2.

Examples

			Solutions (a_1, a_2, a_3) to the equation a_1^2 + 2*a_2^2 + 3*a_3^2 = 9.
------------------------------------------------------------------------
( 1,  2,  0), ( 1, -2,  0),
(-1,  2,  0), (-1, -2,  0),
( 2,  1,  1), ( 2,  1, -1),
( 2, -1,  1), ( 2, -1, -1),
(-2,  1,  1), (-2,  1, -1),
(-2, -1,  1), (-2, -1, -1),
( 3,  0,  0), (-3,  0,  0).
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (first 91 terms from Seiichi Manyama)

Crossrefs

Cf. A000122, A000290, A320067, A320931, A321139.

Programs

Mathematica

nmax = 20; Table[SeriesCoefficient[Product[EllipticTheta[3, 0, x^k], {k, 1, n}], {x, 0, n^2}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)

PARI

{a(n) = polcoeff(prod(i=1, n, 1+2*sum(j=1, sqrtint(n^2\i), x^(i*j^2)+x*O(x^(n^2)))), n^2)}

Formula

a(n) ~ c * d^n / n^(7/4), where d = 6.8137220913147... and c = 0.178176349247... - Vaclav Kotesovec, Oct 30 2018